Paper · Dec 15, 2021

On cusps of caustics by reflection: a billiard variation on Jacobi's Last Geometric Statement

Gil Bor, Serge Tabachnikov

math.DGWider flowsarXiv:2112.07852v2
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Abstract

A point source of light is placed inside an oval. The -th caustic by reflection is the envelope of the light rays emanating from the light source after reflections off the curve. We show that each of these caustics, for a generic point light source, has at least 4 cusps. This is a billiard variation on Jacobi's Last Geometric Statement, concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.

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Submitted
Dec 15, 2021
Last revised
Dec 24, 2021 (v2)
Subjects
math.DG, math.DS, math.SG

Comments: Minor changes in text and figures

Curve shortening

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)1Peachey 2022 Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow (Gabriel Khan, 2023)7Khan 2023 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)4Sobnack 2024 Curve Shortening Flow of Space Curves with Convex Projections (Qi Sun, 2024)2Sun 2024 Curve shortening flow with an ambient force field (Sam Cuthbertson, Glen Wheeler and Valentina-Mira Wheeler, 2024)5Cuthbertson 2024 A delayed interior area-to-height estimate for the Curve Shortening Flow (Arjun Sobnack, 2025)6Sobnack 2025 Huisken's Distance Comparison Principle in Higher Codimension (Qi Sun, 2025)3Sun 2025 Constructing λ-Angenent curve by flow method (Pak Tung Ho, 2026)8Ho 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

  2. Curve Shortening Flow of Space Curves with Convex Projections

    Qi SunWider flows

    Topics Curve shortening

  3. Huisken's Distance Comparison Principle in Higher Codimension

    Qi SunWider flows

    Topics Curve shortening · Shared terms “reflection”

  4. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows

    Topics Curve shortening

  5. Curve shortening flow with an ambient force field

    Sam Cuthbertson, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Curve shortening

  6. A delayed interior area-to-height estimate for the Curve Shortening Flow

    Arjun SobnackWider flows

    Topics Curve shortening

  7. Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow

    Gabriel KhanWider flows

    Topics Curve shortening

  8. Constructing -Angenent curve by flow method

    Pak Tung HoWider flows

    Topics Curve shortening

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  2. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.